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Schrodinger equation that is proportional to the original time: = t.

Problem b: How should the proportionality constant be chosen so that (15.15) trans-

form to:

h2

@G(r ) 2 G(r ) = C (r) ( ) : (15.16)

@ 2m

; r

CHAPTER 15. GREEN'S FUNCTIONS, EXAMPLES

196

The constant C in the right hand side cannot easily be determined from the change

of variables = t because is not necessarily real and it is not clear how a delta

function with a complex argument should be interpreted. For this reason we will

bother to specify C.

The key point to note is that this equation is of exactly the same form as the heat equation

(15.1), where h2 =2m plays the role of the heat conductivity . The only di erence is the

constant C in the right hand side of (15.16). However, since the equation is linear, this

term only leads to an overall multiplication with C.

Problem c: Show that the Green's function for the Green's function can be obtained

from the Green's function (15.14) for the heat equation by making the following

substitutions:

t it=h

;!

h2 =2m (15.17)

;!

G CG

;!

It is interesting to note that the \di usion constant" that governs the spreading of the

waves with time is proportional to the square of Planck's constant. Classical mechanics

follows from quantum mechanics by letting Planck's constant go to zero: h 0. It follows

!

from (15.17) that in that limit the di usion constant of the matter waves goes to zero. This

re ects the fact that in classical mechanics the probability of the presence for a particle

does not spread-out with time.

Problem d: Use the substitutions (15.17) to show that the Green's function for the

Schrodinger equation in N-dimensions is given by:

1 2

G(r t) = C N=2 exp imr =2ht : (15.18)

(2 iht=m)

This Green's function plays a crucial role in the formulation of the Feynman path

integrals that have been a breakthrough both within quantum mechanics as well as in

other elds. A very clear description of the Feynman path integrals is given by Feynman

and Hibbs 22].

Problem e: Sketch the real part of the exponential exp ;imr2=2ht in the Green's func-

tion for a xed time as a function of radius r. Does the wavelength of the Green's

function increase or decrease with distance?

The Green's function (15.18) actually has an interesting physical meaning which is based

on the fact that it describes the propagation of matter waves injected at t = 0 in the

origin. The Green's function can be written as G = C (2 iht=m);N=2 exp (i ), where the

phase of the Green's function is given by

mr2 :

= (15.19)

2ht

As you noted in problem e the wave-number of the waves depends on position. For a

plane wave exp(ik r) the phase is given by = (k r) and the wave-number follows by

taking the gradient of this function.

15.3. THE HELMHOLTZ EQUATION IN 1,2,3 DIMENSIONS 197

Problem f: Show that for a plane wave that

k=r : (15.20)

The relation (15.20) has a wider applicability than plane waves. It is shown by Whitham 66]

that for a general phase function (r) that varies smoothly with r the local wave-number

k(r) is de ned by (15.20).

Problem g: Use this to show that for the Green's function of the Schrodinger equation

the local wave-number is given by

k = mr : (15.21)

ht

Problem h: Show that this expression is equivalent to expression (6.19) of section (6.4):

v = hk (6:19) again

m

In problem e you discovered that for a xed time, the wavelength of the waves decreases

when the distance r to the source is increased. This is consistent with expression (6.19)

when a particle has moved further away form the source in a xed time, its velocity is

larger. This corresponds according to (6.19) with a larger wave-number and hence with a

smaller wavelength. This is indeed the behavior that is exhibited by the full wave function

(15.18).

The analysis of this chapter was not very rigorous because the substitution t (i=h) t

!

implies that the independent parameter is purely imaginary rather than real. This means

that all the arguments used in the previous section for the complex integration should be

carefully re-examined. However, a more rigorous analysis shows that (15.18) is indeed the

correct Green's function for the Schrodinger equation ?]. However, the approach taken

in this section shows that an educated guess can be very useful in deriving new results.

One can in fact argue that many innovations in mathematical physics have been obtained

using intuition or analogies rather than formal derivations. Of course, a formal derivation

should ultimately substantiate the results obtained from a more intuitive approach.

15.3 The Helmholtz equation in 1,2,3 dimensions

The Helmholtz equation plays an important role in mathematical physics because it is

closely related to the wave equation. A very complete analysis of the Green's function

for the wave equation and the Helmholtz equation in di erent dimensions is given by

DeSanto ?]. The Green's function for the wave equation for a medium with constant

velocity c satis es:

t r0 t0 ) c12 @ G(r@t2 r0 t0 ) = (r r0 ) (t t0 ) :

2 t

2 G(r (15.22)

r ; ; ;

As shown in section (15.1) the Green's function depends only on the relative location

r r0 and the relative time t;t0 so that without loss of generality we can take the source

;

CHAPTER 15. GREEN'S FUNCTIONS, EXAMPLES

198

at the origin (r0 = 0) and let the source act at time t0 = 0. In addition it follows from

symmetry considerations that the Green's function depends only on the relative distance

r0 but not on the orientation of the vector r r0 . This means that the Green's

jr ; j ;

_

function then satis es G(r t r0 t0 ) = G(jr r0 t;t0 ) and we need to solve the following

; j

equation:

2

2 G(r t) 1 @ G(r t) = (r) (t) : (15.23)

c2 @t2

r ;

Problem a: Under which conditions is this approach justi ed?

Problem b: Use a similar treatment as in section (15.1) to show that when the Fourier

transform (11.43) is used the Green's function satis es in the frequency domain the

following equation:

2 G(r !) + k2 G(r !) = (r) (15.24)

r

where the wave number k satis es k = !=c.

This equation is called the Helmholtz equation, it is the reformulation of the wave equation

in the frequency domain. In the following we will suppress the factor ! inthe Green's

function but it should be remembered that the Green's function depends on frequency.

We will solve (15.24) for 1,2 and 3 space dimensions. To do this we will consider the

case of N-dimensions and derive the Laplacian of a function F(r) that depends only on

qPN 2

the distance r = j=1 xj . According to expression (4.19) @r=@xj = xj =r. This means

that the derivative @F=@xj can be written as @F=@xj = (@r=@xj ) @F=@r = (xj =r) @F=@r.

Problem c: Use these results to show that

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